Properties

Degree 4
Conductor $ 2^{9} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·7-s − 6·9-s + 8·11-s − 4·13-s + 3·25-s − 16·31-s − 8·35-s − 16·43-s − 12·45-s + 8·47-s + 9·49-s + 16·55-s − 4·61-s + 24·63-s − 8·65-s + 16·67-s − 32·77-s + 27·81-s + 16·91-s − 48·99-s + 12·101-s + 8·103-s + 36·113-s + 24·117-s + 26·121-s + 4·125-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s − 2·9-s + 2.41·11-s − 1.10·13-s + 3/5·25-s − 2.87·31-s − 1.35·35-s − 2.43·43-s − 1.78·45-s + 1.16·47-s + 9/7·49-s + 2.15·55-s − 0.512·61-s + 3.02·63-s − 0.992·65-s + 1.95·67-s − 3.64·77-s + 3·81-s + 1.67·91-s − 4.82·99-s + 1.19·101-s + 0.788·103-s + 3.38·113-s + 2.21·117-s + 2.36·121-s + 0.357·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(627200\)    =    \(2^{9} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{627200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 627200,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.177809811\)
\(L(\frac12)\)  \(\approx\)  \(1.177809811\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.629421562945557860194185813333, −8.039509248603006371759825232792, −7.20010298760058917622488407443, −6.97238544391138654785050907556, −6.51706798292895129929791659218, −6.13212826893626015368391224991, −5.70093888866808737514223874840, −5.38082200571879524884664388449, −4.72005309487925477696355519235, −3.89460518241716095379897881539, −3.29952754400341080064992418043, −3.26180587408034592610487247010, −2.24932759963578088804921436119, −1.83389798555371103662117093708, −0.52188147591422961956475857875, 0.52188147591422961956475857875, 1.83389798555371103662117093708, 2.24932759963578088804921436119, 3.26180587408034592610487247010, 3.29952754400341080064992418043, 3.89460518241716095379897881539, 4.72005309487925477696355519235, 5.38082200571879524884664388449, 5.70093888866808737514223874840, 6.13212826893626015368391224991, 6.51706798292895129929791659218, 6.97238544391138654785050907556, 7.20010298760058917622488407443, 8.039509248603006371759825232792, 8.629421562945557860194185813333

Graph of the $Z$-function along the critical line